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58 IEEE TRANSACTIONS ON EVOLUTIONARY COMPUTATION, VOL. 6, NO. 1, FEBRUARY 2002
The Particle SwarmExplosion, Stability, andConvergence in a Multidimensional Complex Space
Maurice Clerc and James Kennedy
AbstractThe particle swarm is an algorithm for finding op-timal regions of complex search spaces through the interaction ofindividuals in a population of particles. Even thoughthe algorithm,which is based on a metaphor of social interaction, has been shownto perform well, researchers have not adequately explained howit works. Further, traditional versions of the algorithm have hadsome undesirable dynamical properties, notably the particles ve-locities needed to be limited in order to control their trajectories.The present paper analyzes a particles trajectory as it moves indiscrete time (the algebraic view), then progresses to the view ofit in continuous time (the analytical view). A five-dimensional de-piction is developed, which describes the system completely. Theseanalyses lead to a generalized model of the algorithm, containing
a set of coefficients to control the systems convergence tendencies.Some results of the particle swarm optimizer, implementing modi-fications derived fromthe analysis,suggest methods for altering theoriginal algorithm in ways that eliminate problems and increasetheability of theparticleswarm to find optimaof some well-studiedtest functions.
Index TermsConvergence, evolutionary computation, opti-mization, particle swarm, stability.
I. INTRODUCTION
PARTICLE swarm adaptation has been shown to suc-cessfully optimize a wide range of continuous functions[1][5]. The algorithm, which is based on a metaphor of social
interaction, searches a space by adjusting the trajectories ofindividual vectors, called particles as they are conceptualized
as moving points in multidimensional space. The individual
particles are drawn stochastically toward the positions of
their own previous best performance and the best previous
performance of their neighbors.
While empirical evidence has accumulated that the algorithm
works, e.g., it is a useful tool for optimization, there has thus
far been little insight into how it works. The present analysis
begins with a highly simplified deterministic version of the par-
ticle swarm in order to provide an understanding about how it
searches the problem space [4], then continues on to analyze
the full stochastic system. A generalized model is proposed, in-
cluding methods for controlling the convergence properties ofthe particle system. Finally, some empirical results are given,
showing the performance of various implementations of the al-
gorithm on a suite of test functions.
Manuscript received January 24, 2000; revised October 30, 2000 and April30, 2001.
M. Clercis withthe France Tlcom, 74988 Annecy, France (e-mail: [email protected]).
J. Kennedy is with the Bureau of Labor Statistics, Washington, DC 20212USA (e-mail: [email protected]).
Publisher Item Identifier S 1089-778X(02)02209-9.
A. The Particle Swarm
A population of particles is initialized with random positions
and velocities and a function is evaluated, using the par-
ticles positional coordinates as input values. Positions and ve-
locities are adjusted and the function evaluated with the new
coordinates at each time step. When a particle discovers a pat-
tern that is better than any it has found previously, it stores thecoordinates in a vector . The difference between (the best
point found by so far) and the individuals current position
is stochastically added to the current velocity, causing the tra-
jectory to oscillate around that point. Further, each particle is
defined within the context of a topological neighborhood com-prising itself and some other particles in the population. The
stochastically weighted difference between the neighborhoods
best position and the individuals current position is also
added to its velocity, adjusting it for the next time step. These
adjustments to the particles movement through the space cause
it to search around the two best positions.
The algorithm in pseudocode follows.
Intialize population
Do
For to Population Size
if then
For to Dimension
0 0
sign 1 abs
Next
Next
Until termination criterion is met
The variables and are random positive numbers, drawn
from a uniform distribution and defined by an upper limit ,
which is a parameter of the system. In this version, the term vari-
able is limited to the range for reasons that will beexplained below. The values of the elements in are deter-
mined by comparing the best performances of all the members
of s topological neighborhood, defined by indexes of some
other population members and assigning the best performers
index to the variable . Thus, represents the best position
found by any member of the neighborhood.
The random weighting of the control parameters in the al-
gorithm results in a kind of explosion or a drunkards walk
as particles velocities and positional coordinates careen toward
infinity. The explosion has traditionally been contained through
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CLERC AND KENNEDY: THE PARTICLE SWARMEXPLOSION, STABILITY, AND CONVERGENCE 59
implementation of a parameter, which limits step size or
velocity. The current paper, however, demonstrates that the im-
plementation of properly defined constriction coefficients can
prevent explosion; further, these coefficients can induce parti-
cles to converge on local optima.
An important source of the swarms search capability is the
interactions among particles as they react to one anothers find-
ings. Analysis of interparticle effects is beyond the scope of thispaper, which focuses on the trajectories of single particles.
B. Simplification of the System
We begin the analysis by stripping the algorithm down to a
most simple form; we will add things back in later. The particle
swarm formula adjusts the velocity by adding two terms to it.
The two terms are of the same form, i.e., , where is
the best position found so far, by the individual particle in the
first term, or by any neighbor in the second term. The formula
can be shortened by redefining as follows:
Thus, we can simplify our initial investigation by looking at
the behavior of a particle whose velocity is adjusted by only one
term
where . This is algebraically identical to the stan-
dard two-term form.
When the particle swarm operates on an optimization
problem, the value of is constantly updated, as the system
evolves toward an optimum. In order to further simplify the
system and make it understandable, we set to a constant
value in the following analysis. The system will also bemore understandable if we make a constant as well; where
normally it is defined as a random number between zero and a
constant upper limit, we will remove the stochastic component
initially and reintroduce it in later sections. The effect of on
the system is very important and much of the present paper is
involved in analyzing its effect on the trajectory of a particle.
The system can be simplified even further by considering a
one-dimensional (1-D) problem space and again further by re-
ducing the population to one particle. Thus, we will begin by
looking at a stripped-down particle by itself, e.g., a population
of one 1-D deterministic particle, with a constant .
Thus, we begin by considering the reduced system
(1.1)
where and are constants. No vector notation is necessary
and there is no randomness.
In [4], Kennedy found that the simplified particles trajectory
is dependent on the value of the control parameter and recog-
nized that randomness was responsible for the explosion of the
system, although the mechanism that caused the explosion was
not understood. Ozcan and Mohan [6], [7] further analyzed the
system and concluded that the particle as seen in discrete time
surfs on an underlying continuous foundation of sine waves.
The present paper analyzes the particle swarm as it moves in
discrete time (the algebraic view), then progresses to the view of
it in continuous time (the analytical view). A five-dimensional
(5-D) depiction is developed, which completely describes the
system. These analyses lead to a generalized model of the al-
gorithm, containing a set of coefficients to control the systems
convergence tendencies. When randomness is reintroduced to
the full model with constriction coefficients, the deleterious ef-fects of randomness are seen to be controlled. Some results of
the particle swarm optimizer, using modifications derived from
the analysis, are presented; these results suggest methods for al-
tering the original algorithm in ways that eliminate some prob-
lems and increase the optimization power of the particle swarm.
II. ALGEBRAICPOINT OFVIEW
The basic simplified dynamic system is defined by
(2.1)
where .Let
be the current point in and
the matrix of the system. In this case, we have
and, more generally, . Thus, the system is defined
completely by .
The eigenvalues of are
(2.2)
We can immediately see that the value is special.
Below, we will see what this implies.For , we can define a matrix so that
(2.3)
(note that does not exist when ).For example, from the canonical form , we find
(2.4)
In order to simplify the formulas, we multiply by to pro-
duce a matrix
(2.5)
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62 IEEE TRANSACTIONS ON EVOLUTIONARY COMPUTATION, VOL. 6, NO. 1, FEBRUARY 2002
The coefficients and depend on and . If
, we have
(3.7)
In the case where , (3.5) and (3.6) give
(3.8)
so we must have
(3.9)
in order to prevent a discontinuity.
Regarding the expressions and , eigenvalues of the ma-
trix , as in Section II above, the same discussion about the
sign of ( ) can be made, particularly about the (non) ex-
istence of cycles.The above results provide a guideline for preventing the ex-
plosion of thesystem, forwe canimmediatelysee that it depends
on whether we have
(3.10)
B. A Posteriori Proof
One can directly verify that and are, indeed, solu-
tions of the initial system.
On one hand, from their expressions
(3.11)
and on the other hand
(3.12)
and also
(3.13)
C. General Implicit and Explicit Representations
A more general implicit representation (IR) is produced by
adding five coefficients , which will allow us to
identify how the coefficients can be chosen in order to ensure
convergence. With these coefficients, the system becomes
(3.14)
The matrix of the system is now
Let and be its eigenvalues.
The (analytic) explicit representation (ER) becomes
(3.15)
with
(3.16)
Now the constriction coefficients (see Section IV for details)
and are defined by
(3.17)
with
(3.18)
which are the eigenvalues of the basic system. By computing
the eigenvalues directly and using (3.17), and are
(3.19)
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64 IEEE TRANSACTIONS ON EVOLUTIONARY COMPUTATION, VOL. 6, NO. 1, FEBRUARY 2002
3) Class Model: A second model related to the Class 1
formula is defined by
(3.31)
(3.32)
For historical reasons and for its simplicity, the case has
been well studied. See Section IV-C for further discussion.
4) Class 2 Model: A second class of models is defined by
the relations
(3.33)
Under these constraints, it is clear that
(3.34)
which gives us and , respectively.
Again, an easy way to obtain real coefficients for every
value is to have . In this case
(3.35)
In the case where , the following is obtained:
(3.36)
From the standpoint of convergence, it is interesting to note
that we have the following.
1) For the Class 1 models, with the condition
(3.37)
2) For the Class models, with the conditions
and
(3.38)3) For the the Class 2 models, see (3.39) at the bottom of the
page, with .
This means that we will just have to choose ,
, and , class , respectively, to have a
convergent system. This will be discussed further in Section IV.
F. Removing the Discontinuity
Depending on the parameters the systemmay have a discontinuity in due to the presence of the term
in the eigen-
values.
Thus, in order to have a completely continuous system, thevalues for must be chosen such that
(3.40)
By computing the discriminant, the last condition is found to
be equivalent to
(3.41)
In order to be physically plausible, the parameters
must be positive. So, the condition becomes
(3.42)The set of conditions taken together specify a volume in
for the admissible values of the parameters.
G. Removing the Imaginary Part
When the condition specified in (3.42) is met, the trajectory
is usually still partly in a complex space whenever one of the
eigenvalues is negative, due to the fact that is a complex
(3.29)
(3.39)
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CLERC AND KENNEDY: THE PARTICLE SWARMEXPLOSION, STABILITY, AND CONVERGENCE 65
number when is not an integer. In order to prevent this, we
must find some stronger conditions in order to maintain positive
eigenvalues.
Since
(3.43)
the following conditions can be used to ensure positive eigen-values:
(3.44)
Note 3.2: From an algebraic point of view, the conditions
described in (3.43) can be written as
trace (3.45)
Now, these conditions depend on . Nevertheless, if the max-
imum value is known, they can be rewritten as
(3.46)
Under these conditions, all system variables are real numbers
in conjunction with the conditions in (3.42) and (3.44), the pa-
rameters can be selected so that the system is completely con-
tinuousandreal.
H. Example
As an example, suppose that and . Now the
conditions become
(3.47)
For example, when
(3.48)
the system converges quite quickly after about 25 time steps
and at each time step the values of and are almost the same
over a large range of values. Fig. 2(a) shows an example ofconvergence ( and ) for a continuous real-valued
system with .
I. Reality and Convergence
The quick convergence seen in the above example suggests
an interesting question. Does realityusing real-valued vari-
ablesimply convergence? In other words, does the following
hold for real-valued system parameters:
(3.49)
(a)
(b)
Fig. 2. (a) Convergent trajectory in phase space of a particle when and , where . Both velocity and , the difference between theprevious best , and the current position converge to 0.0. (b) increases overtime, even when the parameters are real and not complex.
The answer is no. It can be demonstrated that convergence is
not always guaranteed for real-valued variables. For example,given the following parameterization:
(3.50)
the relations are
(3.51)
which will produce system divergence when (for in-
stance), since . This is seen in Fig. 2(b)
IV. CONVERGENCE ANDSPACE OFSTATES
From the general ER, we find the criterion of convergence
(4.1)
where and are usually true complex numbers.
Thus, the whole system can be represented in a 5-D space
Re Im Re Im .
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CLERC AND KENNEDY: THE PARTICLE SWARMEXPLOSION, STABILITY, AND CONVERGENCE 67
or
(4.12)
Thus
(4.13)
with
trace determinant
(4.14)
Step 3) Complex and Real Areas onThe discriminant is negative for the values in
. In
this area, the eigenvalues are true complex numbersand their absolute value (i.e., module) is simply .
Step 4) Extension of the Complex Region and Constriction
Coefficient
In the complex region, according to the conver-
gence criterion, in order to get convergence.
So the idea is to find a constriction coefficient de-
pending on so that the eigenvalues are true com-
plex numbers for a large field of values. In this
case, the common absolute value of the eigenvalues
is
for
else
(4.15)
which is smaller than one for all values as soon as
is itself smaller than one.
This is generally the most difficult step and sometimes needs
some intuition. Three pieces of information help us here:
1) the determinant of the matrix is equal to ;
2) this is the same as in Constriction Type 1;
3) we know from the algebraic point of view the system is
(eventually) convergent like .
So it appears very probable that the same constriction coeffi-
cient used for Type 1 will work. First, we try
(4.16)
that is to say
for
else
(4.17)
It is easy to see that is negative only between and ,
depending on . The general algebraic form of is quite
complicated (polynomial in with some coefficients being
roots of an equation in ) so it is much easier to compute
it indirectly for some values. If is smaller than four,
then and by solving we find that
Fig. 4. Discriminant remains negative within some bounds of , dependingon the value of , ensuring that the particle system will eventually converge.
TABLE IIVALUES OF BETWEENWHICH THEDISCRIMINANTIS NEGATIVE,
FORTWOSELECTEDVALUES OF
. This relation is valid as soon as .
Fig. 4 shows how the discriminant depends on , for two
values. It is negative between the values given in Table II.
D. Moderate Constriction
While it is desirable for the particles trajectory to converge,
by relaxing the constriction the particle is allowed to oscillate
through the problem space initially, searching for improvement.
Therefore, it is desirable to constrict the system moderately,
preventing explosion while still allowing for exploration.
To demonstrate how to produce moderate constriction, the
following ER is used:
(4.18)
that is to say
From the relations between ER and IR, (4.19) is obtained, as
shown at the bottom of the next page.
There is still an infinity of possibilities for selecting the pa-
rameters . In other words, there are many different IRs
that produce the same explicit one. For example
(4.20)
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68 IEEE TRANSACTIONS ON EVOLUTIONARY COMPUTATION, VOL. 6, NO. 1, FEBRUARY 2002
Fig. 5. Real parts of and , varying over 50 units of time, for a range of values.
or
(4.21)
From a mathematical point of view, this case is richer than
the previous ones. There is no more explosion, but there is not
always convergence either. This system is stabilized in the
sense that the representative point in the state space tends to
move along an attractor which is not always reduced to a single
point as in classical convergence.
E. Attractors and Convergence
Fig. 5 shows a three-dimensional representation of the
real restriction Re Re of a particle moving in
the 5-D space. Fig. 6(a)(c) show the real restrictions(Re Re ) of the particles that are typically studied. We
can clearly see the three cases:
1) spiral easy convergence toward a nontrivial attractor for
[see Fig. 6(a)];
2) difficult convergence for [see Fig. 6(b)];
3) quick almost linearconvergence for [seeFig. 6(c)].
Nevertheless, it is interesting to have a look at the true system,
including the complex dimensions. Fig. 6(d)(f) shows some
other sections of the whole surface in .
Note 4.2: There is a discontinuity, for the radius is equal
to zero for (see Fig. 7).
Thus, what seems to be an oscillation in the real space is infact a continuous spiralic movement in a complex space. More
importantly, the attractor is very easy to define: it is the circle
[center (0,0) and radius ]. When , and
when , then ( with ), for
the constriction coefficient has been precisely chosen so that
(a) (b)
(c) (d)
(e) (f)
Fig. 6. Trajectories of a particle in phase space with three different values of . (a)(c) and(e) Real parts of the velocity and position relative to the previousbest . (b)(d) and(f) Real andimaginary parts of .(a) and(d) show theattractorfor a particle with . Particle tends to orbit, rather than converging to0.0. (b) and (e) show the same views with . (c) and (f) depict the
easy convergence toward 0.0 of a constricted particle with
. Particleoscillates with quickly decaying amplitude toward a point in the phase space(and the search space).
the part of tends to zero. This provides an intu-
itive way to transform this stabilization into a true convergence.
(4.19)
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CLERC AND KENNEDY: THE PARTICLE SWARMEXPLOSION, STABILITY, AND CONVERGENCE 69
(a)
(b)
Fig. 7. Trumpet global attractor when . Axis Re Im . (a) Effect on of the real and imaginary parts of . (b) Effects of the real and
imaginary parts of .
We just have to use a second coefficient in order to reduce the
attractor, in the case , so that
(4.22)
The models studied here have only one constriction coeffi-
cient. If one sets , the Type 1 constriction is produced,
but now, we understand better why it works.
V. GENERALIZATION OF THEPARTICLE-SWARMSYSTEM
Thus far, the focus has been on a special version of the particle
swarm system, a system reduced to scalars, collapsed terms and
nonprobabilistic behavior. The analytic findings can easily be
generalized to the more usual case where is random and two
vector terms are added to the velocity. In this section the results
are generalized back to the original system as defined by
(5.1)
Now , , and are defined to be
(5.2)
to obtain exactly the original nonrandom system described in
Section I.
For instance, if there is a cycle for , then there is an
infinity of cycles for the values so that .
Upon computing the constriction coefficient, the following
form is obtained:
if
else
(5.3)
Coming back to the ( ) system, and are
(5.4)The use of the constriction coefficient can be viewed as a rec-
ommendation to the particle to take smaller steps. The conver-
gence is toward the point (
). Remember is in fact the velocity of the particle, so it will
indeed be equal to zero in a convergence point.2 Example
and are uniform random variables between 0 and
and respectively. This example is shown in Fig. 8.
VI. RUNNING THEPARTICLE SWARMWITHCONSTRICTION
COEFFICIENTS
As a resultof theabove analysis, the particle swarm algorithm
can be conceived of in such a way that the systems explosion
can be controlled, without resorting to the definition of any ar-
bitrary or problem-specific parameters. Not only can explosion
be prevented, but the model can be parameterized in such a way
that the particle system consistently converges on local optima.
(Except for a special class of functions, convergence on global
optima cannot be proven.)
The particle swarm algorithm can now be extended to include
many types of constriction coefficients. The most general mod-ification of the algorithm for minimization is presented in the
following pseudocode.
Assign
Calculate
Initialize population: random
Do
For to population size
2Convergence implies velocity , but the convergent point is not neces-sarily the one we want, particularly if the system is tooconstricted. We hope toshow in a later paper how to cope with this problem, by defining the optimalparameters.
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Fig. 8. Example of the trajectory of a particle with the original formulacontaining two 0 terms, where is the upper limit of a uniform randomvariable. As can be seen, velocity converges to 0.0 and the particles position converges on the previous best point .
if then
For to dimension
rand 2
rand 2
0
0
0
0
Next d
Next i
Until termination criterion is met.
In this generalized version of the algorithm, the user selects
the version and chooses values for and that are consistent
with it. Then the two eigenvalues are computed and the greater
one is taken. This operation can be performed as follows.
discrim 0
0
0
a 0
if discrim then
neprim abs
discrim
neprim abs 0
discrim
else
neprim abs discrim
neprim neprim
eig. neprim neprim
These steps are taken only once in each program and, thus, do
not slow it down. For the versions tested in this paper, the con-
striction coefficient is calculated simply as eig. .
For instance, the Type 1 version is defined by the rules
.
The generalized description allows the user to control the de-
gree of convergence by setting to various values. For instance,
in the Type version, results in slow convergence,
meaning that the space is thoroughly searched before the popu-
lation collapses into a point.
In fact, the Type constriction particle swarm can be pro-
grammed as a very simple modification to the standard version
presented in Section I. The constriction coefficient is calcu-
lated as shown in (4.15)
, for
else
The coefficient is then applied to the right side of the velocity
adjustment.
Calculate
Initialize population
Do
For to Population Size
if then
For to Dimension
0
0
Next
Next
Until termination criterion is met.
Note that the algorithm now requires no explicit limit .
The constriction coefficient makes it unnecessary. In [8], Eber-
hart and Shi recommended, based on their experiments, that a
liberal , for instance, one thatis equal to the dynamic rangeof the variable, be used in conjunction with the Type con-
striction coefficient. Though this extra parameter may enhance
performance, the algorithm will still run to convergence even if
it is omitted.
VII. EMPIRICALRESULTS
Several types of particle swarms were used to optimize a set
of unconstrained real-valued benchmark functions, namely, sev-
eral of De Jongs functions [9], Schaffers f6, and the Griewank,
Rosenbrock, and Rastrigin functions. A population of 20 parti-
cles was run for 20 trials per function, with the best performance
evaluation recorded after 2000 iterations. Some results from An-gelines [1] runs using an evolutionary algorithm are shown for
comparison.
Though these functions are commonly used as benchmark
functions for comparing algorithms, different versions have ap-
pearedin the literature. The formulas used here forDe Jongs f1,
f2, f4 (without noise), f5, and Rastrigin functions are taken from
[10]. Schaffers f6 function is taken from [11]. Note that earlier
editions give a somewhat different formula. The Griewank func-
tion given here is the one used in the First International Contest
on Evolutionary Optimization held at ICEC 96 and the 30-di-
mensional generalized Rosenbrock function is taken from [1].
Functions are given in Table III.
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CLERC AND KENNEDY: THE PARTICLE SWARMEXPLOSION, STABILITY, AND CONVERGENCE 71
TABLE IIIFUNCTIONSUSED TOTEST THEEFFECTS OF THE CONSTRICTIONCOEFFICIENTS
TABLE IVFUNCTIONPARAMETERS FOR THETESTPROBLEMS
A. Algorithm Variations Used
Three variations of the generalized particle swarm were usedon the problem suite.
Type 1:The first version applied the constriction coefficientto all terms of the formula
using .Type 1 : The second version tested was a simple constriction,
which was not designed to converge, but not to explode, either,as was assigned a value of 1.0. The model was defined as
Experimental Version:The third version tested was more ex-perimental in nature. The constriction coefficient was initiallydefined as . If , then it was multipliedby 0.9 iteratively. Once a satisfactory value was found, the fol-lowing model was implemented:
As in the first version, a generic value of was used.Table IV displays the problem-specific parameters implementedin the experimental trials.
B. Results
Table V compares various constricted particle swarms per-
formance to that of the traditional particle swarm and evo-
lutionary optimization (EO) results reported by [1]. All particle
swarm populations comprised 20 individuals.
Functions were implemented in 30 dimensions except for f2,
f5, and f6, which are given for two dimensions. In all cases ex-
cept f5, the globally optimal function result is 0.0. For f5, the
best known result is 0.998004. The limit of the control param-
eter was set to 4.1 for the constricted versions and 4.0 for the
versionsof the particle swarm. The columnlabeled E&S
was programmed according to the recommendations of [8]. This
condition included both Type constriction and , with
setto therangeof theinitial domainfor thefunction. Func-
tion results were saved with six decimal places of precision.
As can be seen, the Type and Type 1 constricted versions
outperformed the versions in almost every case; the exper-
imental version was sometimes better, sometimes not. Further,
the Type and Type 1 constricted particle swarms performed
better than the comparison evolutionary method on three of the
four functions. With some caution, we can at least consider the
performances to be comparable.
Eberhart and Shis suggestion to hedge the search by re-
taining with Type constriction does seem to result in
good performance on all functions. It is the best on all except the
Rosenbrock function, where performance was still respectable.An analysis of variance was performed comparing the E&S
version with Type , standardizing data within functions.
It was found that the algorithm had a significant main effect
, , but that there was a significant
interaction of algorithm with function ,
, suggesting that the gain may not be robust across
all problems. These results support those of [8].
Any comparison with Angelines evolutionary method
should be considered cautiously. The comparison is offered
only as aprima faciestandard by which to assess performances
on these functions after this number of iterations. There are
numerous versions of the functions reported in the literature
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[5] Y. Shi and R. C. Eberhart, Parameter selection in particle swarm adap-tation, in Evolutionary Programming VII, V. W. Porto, N. Saravanan,D. Waagen, and A. E. Eiben, Eds. Berlin, Germany: Springer-Verlag,1997, pp. 591600.
[6] E. Ozcan and C. K. Mohanet al., Analysis of a simple particle swarmoptimization problem, in Proc. Conf. Artificial Neural Networks in
Engineering, C. Dagli et al., Eds., St. Louis, MO, Nov. 1998, pp.253258.
[7] , Particle swarm optimization: Surfing the waves, inProc. 1999
Congr. Evolutionary Computation, Washington, DC, July 1999, pp.19391944.
[8] R. C. Eberhart and Y. Shi, Comparing inertia weights and constrictionfactors in particle swarm optimization, in Proc. 2000 Congr. Evolu-tionary Computation, San Diego, CA, July 2000, pp. 8488.
[9] K. De Jong, An analysis of the behavior of a class of genetic adaptivesystems, Ph.D. dissertation, Dept. Comput. Sci., Univ. Michigan, AnnArbor, MI, 1975.
[10] R. G. Reynolds and C.-J. Chung, Knowledge-based self-adaptation inevolutionary programming using cultural algorithms, in Proc. IEEE
Int. Conf. Evolutionary Computation, Indianapolis, IN, Apr. 1997, pp.7176.
[11] L. Davis, Ed.,Handbook of Genetic Algorithms. New York: Van Nos-trand Reinhold, 1991.
Maurice Clerc received the M.S. degree in mathe-matics(algebraand complex functions)from the Uni-versit de Villeneuve, France, and the Eng. degree incomputer science fromthe Institut industrieldu Nord,Villeneuve dAsq, France, in 1972.
He is currently with Research and Design, FranceTlcom, Annecy, France. His current research in-terests include cognitive science, nonclassical logics,and artificial intelligence.
Mr. Clerc is a Member of the French Associationfor Artificial Intelligence and the Internet Society.
James Kennedy received the Masters degree inpsychology from the California State University,Fresno, in 1990 and the Doctorate from the Univer-sity of North Carolina, Chapel Hill, in 1992.
He is currently a Social Psychologist with the Bu-reau of Labor Statistics, Washington, DC, working indata collection research. He has been working withparticle swarms since 1994.